Finite Group Databasebeta

← #38767833

Subgroups — #38767833

Order 25472.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27164608C211yesmaximal, characteristic
2#27164173C211yesmaximal, characteristic, complemented
3#27164159C211yesmaximal, characteristic, complemented
4#27163931C211yesmaximal, characteristic, complemented
5#27163751C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic, complemented
7#328×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C2×#15691876—21no—
10C2×#15691875—21no—
11#20263950—21no—
12#20263944—21no—
13#20263940—21no—
14#20263930—21no—
15#20263929C2211yescharacteristic
16#20263925—21no—
17(C22⋊D199)⋊C4—21no—
18C2×(C2×D199)⋊C4—21no—
19C2×(C2×D199)⋊C4C2211yescharacteristic
20#20263880—21no—
21#20263878—21no—
22C22⋊(C199⋊C8)C2211yescharacteristic
23C199⋊#95—21no—
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398—42no—
31C2×D796—21no—
32C2×D796—21no—
33C2×D796D411yescharacteristic
34C2×C22⋊D199—21no—
35C22×Dic199—21no—
36C22×Dic199D411yescharacteristic
37C2×C4×D199—21no—
38#15691876—42no—
39#15691875—42no—
40C2×C199⋊C8—21no—
41C4⋊Dic199D411yescharacteristic
42Dic199⋊C4—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—21no—
48(C2×D199)⋊C4—21no—
49(C2×D199)⋊C4—42no—
50C4×Dic199—21no—
51C2×Q8×C199D411yescharacteristic
52C22×C4×C199—21no—
53C22×C4×C199C2311yescharacteristic
54C2×C8×C199—21no—
55C4⋊C4×C199—21no—
56C4⋊C4×C199D411yescharacteristic
57C22⋊C4×C199—21no—
58C22×D199—21no—
59C22×D199—21no—
60C22×D199—21no—
61C22×D199—21no—
62C22×D199—41no—
63C22×D199—41no—
64Dic398—22no—
65Dic398—22no—
66Dic398—41no—
67Dic398—41no—
68D796—41no—
69D796—81no—
70D796—81no—
71C22⋊D199—41no—
72C22⋊D199—41no—
73C2×Dic199—22no—
74C2×Dic199—21no—
75C2×Dic199—21no—
76C2×Dic199—41no—
77C2×Dic199—41no—
78C2×Dic199—41no—
79C4×D199—81no—
80C199⋊C8—41no—
81C23×C199C2×D411yescharacteristic
82Q8×C199—41no—
83C2×C4×C199—21no—
84C2×C4×C199—21no—
85C2×C4×C199—41no—
86C2×C4×C199—41no—
87C2×C4×C199—41no—
88C2×C4×C199—41no—
89C2×C4×C199C2×D411yescharacteristic
90C2×C4×C199C2×D411yescharacteristic
91C8×C199—41no—
92C22×C199—21no—
93C22×C199—21no—
94C22×C199#12011yescharacteristic
95C22×C199#13511yescharacteristic
96C22×C199#13711yescharacteristic
97C2×D199—42no—
98C2×D199—41no—
99C2×D199—41no—
100C2×D199—41no—
101C2×D199—41no—
102C2×D199—41no—
103C2×D199—41no—
104C2×D199—81no—
105C4×C199—21no—
106C4×C199—21no—
107C4×C199—41no—
108C4×C199—41no—
109C4×C199—81no—
110Dic199—41no—
111Dic199—41no—
112Dic199—42no—
113C2×C199—21no—
114C2×C199—21no—
115C2×C199#45211yescharacteristic
116C2×C199#45511yescharacteristic
117C2×C199B(4,𝔽2)11yescharacteristic
118D199—81no—
119D199—81no—
120C199#160611yescharacteristic, complemented, Sylow
121#1606—1991nomaximal, Sylow
122#538—1991no—
123#451—1991no—
124#448—1991no—
125#393—1991no—
126#328D19911yescharacteristic, complemented
127#327—1991no—
128#323—1991no—
129C22×Q8—1991no—
130C2×Q16—3981no—
131C2×SD16—3981no—
132#126—3981no—
133#124—3981no—
134#123—3981no—
135C22⋊Q8C2×D19911yescharacteristic
136#121—3981no—
137#121—1991no—
138#120—3981no—
139C4×D4—3981no—
140C2×C22⋊C4—3981no—
141C2×C22⋊C4—1991no—
142#103—3981no—
143#103—3981no—
144C22⋊C8—1991no—
145C22⋊C8C2×D19911yescharacteristic
146#95—3981no—
147#95C2×D19911yescharacteristic
148C24—1991no—
149C2×Q8—7962no—
150C2×Q8—3981no—
151C2×Q8—3981no—
152C2×Q8D79611yescharacteristic
153C2×D4—3981no—
154C2×D4—3981no—
155C2×D4—3981no—
156C2×D4—1991no—
157C22×C4—21no—
158C22×C4—3981no—
159C22×C4—3981no—
160C22×C4—1991no—
161C22×C4C22×D19911yescharacteristic
162Q16—7962no—
163SD16—7962no—
164C2×C8—21no—
165C2×C8—3981no—
166C4⋊C4—21no—
167C4⋊C4—3981no—
168C4⋊C4—1991no—
169C4⋊C4D79611yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—7962no—
172C22⋊C4—3981no—
173C22⋊C4—3981no—
174C22⋊C4—3981no—
175C22⋊C4—3981no—
176C22⋊C4—3981no—
177C22⋊C4—3981no—
178C42—3981no—
179C23—7961no—
180C23—7961no—
181C23—3981no—
182C23—3981no—
183C23—3981no—
184C23—3981no—
185C23D4×D19911yescharacteristic
186Q8—41no—
187Q8—7961no—
188Q8—7961no—
189Q8—3982no—
190Q8—3982no—
191D4—7961no—
192D4—7961no—
193D4—7961no—
194D4—15921no—
195D4—15921no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—7961no—
203C2×C4—7961no—
204C2×C4—7961no—
205C2×C4—15921no—
206C2×C4—3982no—
207C2×C4—3981no—
208C2×C4—3981no—
209C2×C4D4×D19911yescharacteristic
210C2×C4C2×D79611yescharacteristic
211C8—41no—
212C8—7961no—
213C22—21no—
214C22—21no—
215C22—7962no—
216C22—7961no—
217C22—7961no—
218C22—7961no—
219C22—7961no—
220C22—7961no—
221C22—7961no—
222C22—15921no—
223C22#2026392511yescharacteristic
224C22#2026397611yescharacteristic
225C22#2026398711yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—7961no—
232C4—7961no—
233C4—7962no—
234C2—21no—
235C2—21no—
236C2—15921no—
237C2—15921no—
238C2#2716418311yescharacteristic
239C2#2716419711yescharacteristic
240C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199#120
prime radical#27164608C2
hypercenter#328D199
hypocenterC199#1606
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #27164173C2semidirect
3. #27164159C2semidirect
4. #27163931C2semidirect
5. #27163751C2semidirect
6. #27163741C2semidirect
7. #328×C199C2semidirect
120. C199#1606semidirect
126. #328D199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#271646082
#271641732
#271641592
#271639312
#271637512
#271637412
#328×C1992
#1606199